Maximal matroids in weak order posets
Maximal matroids in weak order posets
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DOI：
10.1016/j.jctb.2023.10.012
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发表时间：
2021-02
期刊：
影响因子：
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通讯作者：
B. Jackson;Shin-ichi Tanigawa
中科院分区：
文献类型：
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作者：
B. Jackson;Shin-ichi Tanigawa
设X是有限集合E的子集族。E上的拟阵称为X-拟阵，如果X中的每个集合都是一个圈。我们开发的技术，确定当存在一个唯一的最大X-拟阵的弱序偏序集的所有X-拟阵在E和制定一个猜想，这将是一个独特的最大拟阵的秩函数时，它存在。该猜想提出了一种新的拟阵秩函数，它推广了极图理论中弱饱和序列的概念。我们验证了各种家庭X的猜想，并表明，如果为真，该猜想可能有重要的应用领域，如组合刚性和低秩矩阵完成。
Let X be a family of subsets of a finite set E. A matroid on E is called an X-matroid if each set in X is a circuit. We develop techniques for determining when there exists a unique maximal X-matroid in the weak order poset of all X-matroids on E and formulate a conjecture which would characterise the rank function of this unique maximal matroid when it exists. The conjecture suggests a new type of matroid rank function which extends the concept of weakly saturated sequences from extremal graph theory. We verify the conjecture for various families X and show that, if true, the conjecture could have important applications in such areas as combinatorial rigidity and low rank matrix completion.